Modeling the HIV Transmission with Reinfection

Transcription

1 Applied Mathematical Sciences, Vol. 9, 2015, no. 87, HIKARI Ltd, Modeling the HIV Transmission with Reinfection Juan C. Castillo Paz, Carlos A. Abello Muñoz and Anibal Muñoz Loaiza Grupo de modelación matemática en epidemiologia (GMME) Maestria en Biomatemáticas, Universidad del Quindío, Armenia Q, Colombia Copyright c 2015 Juan C. Castillo Paz, Carlos A. Abello Muñoz and Anibal Muñoz Loaiza. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract We set a mathematical model for the dynamics of the HIV transmission in a sexually active population, considering infected population and reinfection by sexual contact within the infected population, modeled with a non-linear differential equations system (qualitative analysis of the system is realized). Moreover, the parameters used in the model are obtained using the least-squares method with HIV data of the Armenia Quindio city (Colombia). Keywords: HIV, Reinfection, Mathematical models, Parameter estimation, Basic Reproduction Number 1 Introduction The human immunodeficiency virus is a group of clinical manifestations that appear as consequence of the immune system depression, because of the HIV infection. It is a terminal disease and considered as a XXI century epidemic. This disease has no cure but it can be controlled with medications [4, 5]. HIV has produced millions of deaths in the whole planet, the most deaths, in those countries without prevention mechanisms. In medical terms HIV is a serious clinical disorder that leaves the human body susceptible to direct attack of opportunistic diseases, this disturbance is characterized by a progressive deterioration of the central nervous system. The infected people can be asymptomatic several months or years, before,

2 4324 Juan C. Castillo Paz other clinical issues may appear, such as tuberculosis, hepatitis, toxoplasmosis, pneumonia, etc [1]. The HIV reinfection has been debated for a long time, do not exist a theoretical reason to support that reinfection is not possible, since the immune system does not have a complete control of the initial infection, moreover, doctors maintain the hope that reinfection not occurs or seldom occurs. In this point of view are based the beliefs of HIV-positive people who think that sexual relations without protection or syringe exchange each other do not involve any risk. Almost all the virologists think that reinfection is probable, but they do not know the clinical consequences at short and long term in HIV-positive people, since reinfection can vary person to person [6]. 2 HIV model with reinfection Taking into account the previous statements, we formulate and analyze a model to describe the HIV transmission dynamics in a sexually active population, with the following assumptions: a) differentiation of the infected and reinfected people, b) reinfection interpreted as a virus mutation and an increase of the viral charge because of the infected people contact, c) death rate of the infected and reinfected people by natural causes. The variables and parameters used in this model are: x(t) average number of susceptible persons to acquire HIV, y(t) average number of infected persons in an asymptomatic stage, y r (t) average number of reinfected persons at time t, Λ the constant increment of the susceptible population, β the transmission probability by contact with an infected and undiagnosed person, κ the successful meetings rate by contact between infected and susceptible persons, σ reinfection probability by contact between infected people, α the mortality rate of infected people associated with the HIV infection, µ natural mortality rate, θ the rate of treated people after a time t, and ω is the reinfected people rate that develops the HIV infection [?, 8]. The following compartmental flowchart represents the transmission dynamics of the HIV (Figure 1):

3 Modeling the HIV transmission with reinfection 4325 Λ x(t) βκyx y(t) θy µx σκy 2 µy y r (t) (µ + ω)y r Figure 1: Transmission dynamics of the HIV with reinfection. The differential equations to model the HIV transmission dynamics in a sexually active population are: dx dt dy dt dy r dt = Λ βκyx µx (1) = βκyx (µ + θ)y σκy 2 (2) = σκy 2 (µ + ω)y r (3) where β, κ, µ, θ, σ, α, ω > 0, Λ R and initial conditions x(0) = x 0, y(0) = y 0, y r (0) = y r0, the region of epidemiological sense is: { Ω = (x, y, y r ) R+ 3 : 0 < N Λ } (4) µ The basic reproduction number R 0 have the following expression R 0 (θ) = βκλ µ(µ + θ) this equation indicates the spectral radius of the following generation matrix in a mathematical way. 2.1 Stability analysis of the equilibrium points The stationary solutions are the points where the system does not have any change, namely, the population variation is zero. In terms of the HIV issue,

4 4326 Juan C. Castillo Paz the susceptible, infected, and reinfected population remain constant at time. Through direct calculations we obtain the following solutions: ( ) Λ E 0 = µ, 0, 0 y E 1 = (x, y, yr) with x = Λ βκy + µ y r = σκ µ + ω (y ) 2 where, using easy algebraic calculations it is showed the positivity of the point E 1. From these equilibrium points we formulate the following stability theorems. Theorem 2.1 (Local stability of the ( infection-free ) equilibrium point) The Λ infection-free equilibrium point E 0 =, 0, 0 of the system (1)-(3) is local µ and asymptotically stable if R 0 < 1. Proof: Evaluating the Jacobian matrix (J) in the equilibrium point E 0 and writing it in terms of R 0, we have µ βκλ J(E 0 ) = µ 0 0 (µ + θ) [R 0 1] (µ + ω) Defining the characteristic polynomial J(E 0 ) λi = 0 gives P (λ) = [(µ + ω) + λ]p 2 (λ) (5) where P 2 (λ) = λ 2 γ 0 λ + δ 0 is the characteristic polynomial of the minor matrix J(E 0 ) 33 and γ 0 = µ + (µ + θ)[r 0 1] is the trace of that matrix. δ 0 = µ(µ + θ)[r 0 1] is the determinant of the minor matrix, using the trace-determinant criterion it is obtained R 0 < 1, γ 0 < 0 and δ 0 > 0. As the eigenvalue is negative, the roots of the P (λ) have a real negative part. So then, the equilibrium point E 0 is local and asymptotically stable. Theorem 2.2 (local stability of the equilibrium point with infection) The equilibrium point with infection (E 1 = (x, y, y r)) of the system (1)-(3) is local and asymptotically stable, if R 0 > 1. Proof: Evaluating the J matrix in the equilibrium point E 1, it is obtained: βκŷ µ βκˆx 0 J(E 1 ) = βκŷ βκˆx (µ + θ) 2σκŷ 0 0 2σκŷ (µ + ω)

5 Modeling the HIV transmission with reinfection 4327 and the characteristic polynomial is J(E 1 ) λi = 0 P (λ) = [(µ + ω) + λ]p 2 (λ) (6) with P 2 (λ) = λ 2 + a 1 λ + a 0 the characteristic polynomial of the minor matrix J(E 1 ) 33 where a 1 = βκŷ + µ βκˆx + (µ + θ) + 2σκŷ and a 0 = β 2 κ 2ˆx ŷ + (βκŷ + µ)((µ + θ) + 2σκŷ βκˆx ): If we want to analyze a 1, it is necessary to analyze only the x y difference since the remaining terms are positive, then y x = y Λ βκy + µ = βκ(y ) 2 + µy Λ βκy + µ The denominator always is positive, the quadratic equation of the numerator could have a positive value if their solutions have a positive term obtained using the quadratic formula. y 1 = µ ± µ 2 + 4βκΛ 2βκ Then, there exist at least one positive solution, where y x > 0 then a 1 > 0. In a similar way we analyze the sign of a 0, where at least there is one positive solution, where 2σy βx > 0 then a 0 > 0 The Routh-Hurwitz criterion is fulfilled for the case k = 2 where a 0 > 0, a 1 > 0 and the third eigenvalue λ = (µ+ω) is negative with ŷ > 0, if R 0 > 1, then, the equilibrium point with infection (E 1 ) is local and asymptotically stable [3]. 3 Estimation of the model parameters (HIV) with reinfection We compute the parameters values for the transmission dynamics of the HIV in a sexually active population adjusted to the previously described model with data of the Secretaria de Salud de Armenia Quindío in a period from 2012 to 2014, using the least-squares method, and the desolve code of the R package. We introduce a new variable in the model A(t), this variable represents the accumulated infected number per month. A(t) includes the number of HIV infected people, plus reinfected people, dead people by natural causes and HIV-infected people until that year. Then, the variation of A(t) depends on y and y r, therefore da = dy + dyr. We use the least-squares method adjusted dt dt dt to A(t, Θ), with Θ the parameters vector, to the number of HIV accumulated

6 4328 Juan C. Castillo Paz cases. For this procedure we use the function implemented in the R package with the method L-BFGS-B, with this method we can indicate the ranges of minimization. Table 1 depict the obtained values for the parameters. parameter estimate parameter estimate parameter estimate Λ βκ µ θ σκ ω Infectados Acumulados tiempo Figure 2: Fitted plots to the data (Green line) and accumulated data (Magenta line with dots). Figure 2 illustrates the best fits to the data and accumulated data. Moreover, in figure 3 it is observed that under the estimated parameters, the infected population remains in low levels, since the infected people are passing to the reinfected stage, therefore the new HIV cases in Armenia Q. increase their viral charge. In epidemiological terms, there is a systematic and constant increment in the HIV infected people. 4 Conclusion Incorporation of the reinfection phenomenon allows to study the HIV in a global way, giving a comprehension of the systematic increase of the infected

7 Modeling the HIV transmission with reinfection Comportamiento de las poblaciones Susceptibles, Infectadas y Reinfectadas del VIH Susceptibles Infectados Reinfectados 500 Poblaciones Tiempo Figure 3: Behavior of the susceptible population (Blue dashed line), infected population (Green line) and reinfected population (Red dotted line). population, that is product of the lack of knowledge of the reinfection phenomenon. The estimation of the parameters give us an approach in the theoretical models to treat real issues and gives information to the health agencies for the application of effective plans about disease evolution. Especially in Armenia Quindío Colombia the reinfection phenomenon is causing a constant increment in the HIV infected population. Acknowledgements. The authors thank M.E. Dalia Marcela Pizza and M.C. J. Guerrero-Sánchez (IFUAP-BUAP) for useful discussions References [1] B. Abram, Revista Panamericana de la Salud, Vol 2 No 6, Washington D.C. (1997), [2] G. Paula A, Q. José R, Un modelo de VIH-SIDA con reinfección, Matemáticas: Enseñanza universitaria, universidad del Valle, Diciembre (2002), Vol 10, [3] L. Perko, Differential equations and dynamical systems, Texts in applied mathematics 7, New York, Springer, (2001). [4] OPS, Qué es el sida?, Ed. Organización Panamericana de la Salud, Washington, D.C. (2005).

8 4330 Juan C. Castillo Paz [5] Project Inform, Constituye la reinfección una preocupación para las personas con VIH?, Información, inspiración y defensa para las personas conviviendo con VIH/SIDA, (2003), Project Inform, Inc., 205, San Fransisco, CA. [6] R. Z. Díaz F, J. Jaimes F, A. Rugeles M, Origen no infeccioso del sida: mito o realidad?. Asociación colombiana de infectología, Vol. 11-4, (2007), [7] S. Cassels, S. Clark, M. Morris, Mathematical Models for HIV Transmission Dynamics, Tools for Social and Behavioral Science Research, J Acquir Immune Defic Syndr, Vol 47, Supplement 1, March 1, (2008). [8] Y. Wang, Y. Zhou, J. Wu, J. Heffernan, Oscillatory viral dynamics in a delayed HIV pathogenesis model, Mathematical Biosciences, Vol 219 (2009), Received: March 11, 2015; Published: June 12, 2015